Application of spherical pseudo-differential operators and spherical wavelets for numerical solutions of the fixed altimetry-gravimetry boundary value problem
Two possible solutions of the fixed AGBVP are discussed. Both are based on spherical wavelets generated by the Abel-Poisson kernel as a scale function. This ensures the harmonicity of spherical wavelets, which is necessary in the representation of the harmonic disturbing potential as n-level multiresolution analysis (MRA).
Using GPS/leveling data, the fixed AGBVP II could be transformed into a Neumann boundary value problem. After that, the solution of Neumann’s problem in terms of spherical wavelets given in Freeden and Schneider (1998) can be used. The approximation (low-pass filter) and detail coefficients (band-pass filter) in the spherical wavelet representation are spatially distributed and they can be used for establishing smoothness (compatibility) conditions along the coastline.
Another numerical solution of AGBVPs with compatibility conditions is possible with the combined use of spherical harmonics and spherical wavelets (Freeden and Windheuser, 1997). All boundary conditions could be presented as spherical pseudo-differential operators. In this case the application of compatibility conditions in terms of pseudo-differential operators becomes more straightforward. The compatibility conditions in Svensson (1988) are given in an implicit generalized form in terms of pseudodifferential operators as projections between Sobolev spaces. The compatibility conditions are presented in an explicit form.
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